Chemistry - A Molecular Science

Chapter 8 Solid Materials

8.5

DENSITY AND PACKING EFFICIENCY Density (d)

is the mass-to-volume ratio of a substance (Equation 8.2).

d =

mass of substancevolume of substance

=

mV

Eq. 8.2

Individual atoms get denser moving from left to

right in a period as their mass increases

and their size decreases, but the densities of

elements in the bulk do not increase in such a

regular fashion because the density of an element also depends upon the efficiency with which it packs in the solid state. The

packing efficiency

(PE

) is the fraction of the volume

(typically expressed as a percent) of the unit cell that is actually occupied by particles. The space that is not occupied by particles is called

void space

. Equation 8.3 gives the

mathematical expression for packing efficiency expressed as a percent.

atom^3

NV

volume of particles

PE =

100% =

100%

volume of unit cell

a

××

Eq. 8.3

N is the number of atoms in the unit cell and

a is the edge length of the cubic unit cell.

Vatom

is the volume of a spherical atom of radius

r, so

Vatom

=

4 /^3

rπ

3. Example 8.4

demonstrates how packing efficiencies are determined. Example 8.4

What is the packing efficiency of

a face-centered cubic unit cell?

We apply Equation 8.3 with N = 4 atoms for

a fcc unit cell. We then use Equation 8.1b for

the fcc unit cell (r = 0.354a) to get r in terms of a.

3

atom^33

4

4

4 (0.354 a

(4)(

r

NV

3

3

PE =

100% =

100% =

aa

)

π

π

××

(^3) )
a
⎛⎞⎜⎟⎝⎠
3
100% = 74%×
Table 8.1
Packing efficiencies (PE), numbers of atoms in the
unit cell (N), and coordination numbers (CN) of cubic lattices
Lattice Type
PE
N
CN
Simple
52%
1
6
Body-centered 68%
2
8
Face-centered 74%
4
12
Note that the a
3 terms cancel, so the packing efficiency depends only on the type of unit
cell and is independent of the atomic ra
dius or the size of the unit cell.
As shown in Example 8.4, the packing efficiency of a unit cell depends only on the
cell type; it is independent of edge length and atomic radius because both cancel in the calculation. Table 8.1 gives the packing efficiencies of the cubic unit cells and coordination numbers of the atoms in them. As might be expected, the packing efficiency increases as the number of atoms in the unit cell and the number of atoms that are packed around each atom (coordination number) increases.
© by
North
Carolina
State
University